Testing lumpability and conditional independence in Markovian models

A common practice in time series analysis is to lump the observations into a smaller set of categories and then fit a stationary Markov model to the lumped sequence. This paper examines conditions under which lumping preserves regular (first order) Markov chains and enhances their detectability by standard methods. Chi-squared tests are developed to decide whether these conditions are satisfied by any specific lumpability hypothesis concerning a time series. These tests are used to screen out ineffective lumping schemes when there is reason to believe the data realize a regular chain or when the sequence is too short to estimate a higher order model. Screening lumpability hypotheses adds credibility to higher order models in the sense that they reflect properties of the raw data as opposed to being artifacts of lumping.